Electrical energy is transmitted using power lines. Power lines include electrical conductors configured to conduct the electrical energy. The electrical conductors are supported or suspended from power line structures similar to a power line structure 100 as described below with resects to FIG. 1. Because power lines are exposed to meteorological elements, power lines may be designed and constructed to withstand potential damages that may be caused by vibrations due to meteorological elements such as wind and/or ice, for example. Due to meteorological elements, a number of undesirable vibration phenomenon may occur, for example, “aeolian” vibration (e.g. torsional conductor movement and string vibration) which can lead to conductor fatigue failures and conductor “galloping.” These undesirable vibration phenomenon may result in: i) contact between multiple conductors or between multiple conductors and overhead ground wires (i.e. shields); ii) conductor failure at support points on power line structures due to vibration induced stress; iii) possible power line structure damage; and iv) excessive conductor sag due to conductor overstressing.
Aeolian vibration is a high-frequency low-amplitude oscillation generated by a low velocity, comparatively steady wind blowing across a conductor. This steady wind creates air vortices or eddies on the lee side of the conductor. These vortices or eddies will detach at regular intervals from the top and bottom area of the conductor (i.e. “vortex shedding”) creating a force on the conductor that is alternately impressed from above and below. If the frequency of the forces (i.e. expected excitation frequency) approximately corresponds to a frequency of a resonant vibration mode for a conductor span (i.e natural frequency of the power line), the conductor will tend to vibrate in many loops in a vertical plane. The frequency of resonant vibration depends mainly on conductor size and wind velocity and is generally between 5 and 100 Hz for wind speeds within the range of 0 to 15 miles per hour. The peak-to-peak vibration amplitudes will cause alternating bending stresses great enough to produce fatigue failure in the conductor strands at the attachment points to the power line structure. Highly tensioned conductors in long spans are particularly subject to vibration fatigue. This vibration is generally more severe in flat open terrain where steady winds are more often encountered.
Conductor galloping (sometimes called dancing), is a phenomenon where power line conductors move or vibrate with large amplitudes. Galloping usually occurs when an unsteady, high or gusty wind blows over a conductor covered by a layer of ice deposited by freezing rain, mist, or sleet. The coating may vary from a very thin glaze on one side to a solid three-inch cover giving the conductor an irregularly shaped profile. Consequently, this ice covering may give the conductor a slightly out-of-round, elliptical, or quasi-airfoil shape. Wind blowing over this irregularly shaped profile results in aerodynamic lift that causes the conductor to gallop. The wind can be anything between 5 to 45 miles-per-hour at an angle to the power line of 10 to 90 degrees. The wind may be unsteady in velocity or direction.
During galloping, conductors oscillate elliptically at frequencies on the order of 1-Hz or less with vertical amplitudes of several feet. Sometimes loops appear, superimposed on one basic loop. Single-loop galloping rarely occurs in spans over 600 to 700 feet. This is fortunate because at would be impractical to provide clearances large enough in long spans to prevent the possibility of contact between phases. In double-loop galloping, the maximum amplitude usually occurs at the quarter span points and is smaller than that resulting from single-loop galloping. There are several measures that can be incorporated at the power line's design stage to reduce potential conductor contacts caused by galloping, such as designing the power line to have shorter spans, or increased phase separation.
In areas where galloping is either historically known to occur or is expected, power line designers should indicate design measures that will minimize galloping and galloping problems, especially conductor contacts. The primary tool for assuring absence of conductor contacts is to superimpose Lissajous ellipses over a structure's scaled diagram to indicate a galloping conductors theoretical path. FIG. 1 shows power line structure 100, a first phase Lissajous ellipse 105, a second phase Lissajous ellipse 110, a third phase Lissajous ellipse 115, a first shield Lissajous ellipse 120, and a second shield Lissajous ellipse 125. Ways to calculate the aforementioned Lissajous ellipses is shown in Table 1.
TABLE 1Eq. 6-6  ϕ  =            tan              -        1              ⁡          (                        p          c                          w          c                    )       Single LoopDouble LoopMajor Axis ‘M’M = 1.25 Si + 1.0Eq. 6-7                                 M          =                      1.0            +                                                            3                  ⁢                                      a                    ⁡                                          (                                              L                        +                                                                              8                            ⁢                                                                                                                  ⁢                                                          S                              i                              2                                                                                                            3                            ⁢                            L                                                                          -                                                  2                          ⁢                                                                                                          ⁢                          a                                                                    )                                                                      8                                                                                              where            ⁢                                                  ⁢            a                    =                                                                      (                                      L                    2                                    )                                2                            -                              S                i                2                                                        Eq. 6-8 Distance ‘B’B = 0.25 SiEq. 6-9B = 0.2 MEq. 6-10Minor Axis ‘D’D = 0.4 MEq. 6-11D = 2√{square root over (M − 1.0)}Eq. 6-12Where:pc = wind load per unit length on iced conductor in lbs/ft. Assume a 2 psf wind.wc = weight per unit length of conductor plus 1/2 in. of radial ice, lbs/ftL = span length in feet.M = major axis of Lissajous ellipses in feet.Si = final sag of conductor with 1/2 in. of radial ice. no wind, at 32° F. in feet.D = minor axis of Lissajous ellipes in feet.B, φ = as defined in figure above
To avoid contact between phase conductors or between phase conductors and shield wires, none of the ellipses (i.e. first phase Lissajous ellipse 105, second phase Lissajous ellipse 110, third phase Lissajous ellipse 115, first shield Lissajous ellipse 120, and second shield Lissajous ellipse 125) should touch one another. However, if galloping is expected to be infrequent and of minimal severity, there may be situations where allowing ellipses to overlap may be the favored design choice when economics are considered.